Quotient Rule (In what order do you choose which to differentiate)

Jani08

1. The problem statement, all variables and given/known data

The Derivative of x-2/(x^2-1)

Can you show me how to get the derivative using the quotient rule and tell me if the order matters? I always thought that the order never mattered until I got a wrong graph because it seems I got the wrong derivative.

2. Relevant equations



3. The attempt at a solution

So which one is correct?
If I plug in 0.1 on each of these one gives me a + answer and the other -

I got -x^2+4x-1/(x^2-1)^2
and
x^2-4x+1/(x^2-1)^2

At first I thought these where the same but then as I was plugging in random point to see if it increased or decreased at the critical points I did not get the same results. It seems that ive used the quotient rule correctly but the answer changes depending on which you choose first to differentiate.

Hurkyl

You can evaluate a compound expression in any order you want, as long as you do it correctly. What do you mean by "order matters"? I have a suspicion you are comparing two different formulas, rather than comparing two different ways to evaluate a single formula.

Jani08

Well you know how we have (f/g)

what is (f/g)' ?

is it f'g-g'f/g^2

or g'f-f'g/g^2

OR does it even matter? I thought it didn't matter until I came to this problem

gb7nash

(f'g-g'f)/g^2

Hurkyl

Well, the two formulas

[tex]\frac{f'g - g'f}{g^2}[/tex]

and

[tex]\frac{g'f - f'g}{g^2}[/tex]

are formally different. Of course, that doesn't mean they are actually unequal -- have you thought about trying to prove/disprove their equality? It would be one of those "prove or disprove the following identity" kind of problems. I guess you have already done the "disproof", since you found a counterexample. Do you remember why you thought it didn't matter?


Incidentally, what you wrote is technically wrong: the meaning of

[tex]f'g - g'f/g^2[/tex]

is

[tex]f'g - \frac{g'f}{g^2}[/tex]

and not

[tex]\frac{f'g - g'f}{g^2}[/tex]

Even if you know what you mean, you really should add parentheses when talking to other people:

[tex](f'g - g'f)/g^2[/tex]

because they might not realize that you are abusing notation. Actually, I would advise using parentheses even when you're writing just for yourself -- I've seen many people make mistakes because they got confused on how things were grouped.

Mark44

Note the parentheses that gb7nash added. When you write x-2/(x^2-1), people would interpret this as x - (2/(x^2 - 1)), which is probably not what you intended.

Also, (f'g-g'f)/g^2 and (g'f - f'g)/g^2 have opposite signs.

vela

I suspect you were confused because the order doesn't matter with the product rule. That's because in the product rule, you're adding the terms, and addition is commutative. If you differentiate f first, you get f'g+g'f, and if you differentiate g first, you get g'f+f'g. Since the order doesn't matter for addition, those two expressions are equal.

With the quotient rule, however, there's a subtraction rather than an addition. As you know, the order does matter for subtraction: a-b isn't the same as b-a. So for the quotient rule, you need to remember that you differentiate f first and g second.

If you can't remember that, just write f/g as f(g^-1) and use the product and chain rules to rederive the quotient rule.

Jani08

Hurkyl thanks for tips, I guess I should type it the same way I type it in my graphing calculator.

Vela you got it spot on, now that I look at it the way of properties it does make sense. Hah I finally got it...after a whole year doing this thing..I was always just "hoping" I had the right order never really tackled it.

Thanks everyone!